CS252 Algorithms
Monday, 7 November 2022

+ Class time plan
- Today: network flows as an abstract graph problem
- Wednesday: a bunch of applications of network flows
- Friday: follow-up questions and the beginning
    of some of my favorite algorithms
- Monday: questions, more cool algorithms
- Wednesday: questions, recap, maybe one more cool algorithm

+ Network flows
- Definitions: flow network, source, sink, capacity
               flow, residual graph, augmenting path
- Max flow algorithm (Ford-Fulkerson)
- Questions: does it terminate?; does it maximize
    the flow? how long does it take?
- Definitions: cut, min cut
- Theorem: max-flow, min-cut

+ Algorithm (Ford-Fulkerson)
Initialize f to be f(e) = 0 for all e in E
Initialize residual graph G_f accordingly
while there exists a path P from s to t in the G_f:
    let "bottleneck" of P be the minimum edge capacity in P
    push bottleneck's worth of flow onto all edges in P
        and update f and G_f accordingly